Question:hard

If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is

Updated On: Jan 15, 2026

  • \(\frac{\sqrt{37}+\sqrt{13}}{2}\)

  • \(\frac{\sqrt{13}+\sqrt{12}}{2}\)

  • \(\sqrt{(37)} + \sqrt{(13)}\)

  • \(\sqrt{(13)} + \sqrt{(12)}\)

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The Correct Option is C

Solution and Explanation

Given:

  • Area of rhombus = 12 cm²
  • Side of the rhombus = 5 cm

Step 1: Define diagonals

Let the longer diagonal be \( 2a \) and the shorter diagonal be \( 2b \).

Step 2: Apply rhombus area formula

The area of a rhombus is \( \frac{1}{2} \cdot d_1 \cdot d_2 \). Substituting the defined diagonals, we get \( \text{Area} = \frac{1}{2} \cdot 2a \cdot 2b = 2ab \). Given the area is 12 cm², we have \( 2ab = 12 \), which simplifies to \( ab = 6 \) (Equation 1).

Step 3: Use Pythagoras Theorem

The diagonals of a rhombus bisect each other at right angles. Therefore, half of each diagonal forms the legs of a right-angled triangle, with the side of the rhombus as the hypotenuse. Applying the Pythagorean theorem, \( \text{Side}^2 = a^2 + b^2 \). Given the side is 5 cm, we have \( 5^2 = a^2 + b^2 \), so \( a^2 + b^2 = 25 \) (Equation 2).

Step 4: Utilize algebraic identities

We use the sum and difference formulas for squares:
\( (a + b)^2 = a^2 + b^2 + 2ab \). Substituting values from Equations 1 and 2: \( (a + b)^2 = 25 + 2 \cdot 6 = 37 \). Thus, \( a + b = \sqrt{37} \) (Equation 3).
\( (a - b)^2 = a^2 + b^2 - 2ab \). Substituting values from Equations 1 and 2: \( (a - b)^2 = 25 - 2 \cdot 6 = 13 \). Thus, \( a - b = \sqrt{13} \) (Equation 4).

Step 5: Solve for \( a \)

Add Equation 3 and Equation 4: \( (a + b) + (a - b) = \sqrt{37} + \sqrt{13} \). This simplifies to \( 2a = \sqrt{37} + \sqrt{13} \). Therefore, \( a = \frac{\sqrt{37} + \sqrt{13}}{2} \). The longer diagonal is \( 2a \), which is \( \sqrt{37} + \sqrt{13} \).

✅ Final Answer: \( \boxed{2a = \sqrt{37} + \sqrt{13}} \)

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